Countable Complement Space is T1

Theorem

Let $T = \struct {S, \tau}$ be a countable complement topology.

Then $T$ is a $T_1$ (Fréchet) space.

Proof

We have that the Countable Complement Topology is Expansion of Finite Complement Topology.

We also have that a Finite Complement Space is $T_1$.

Then from Separation Properties Preserved by Expansion, we have that $T$ is a $T_1$ (Fréchet) space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $20$. Countable Complement Topology: $1$